Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.
Investigates portfolio selection for rank-dependent utilities in incomplete markets.
problem Portfolio selection for agents with rank-dependent utility in incomplete financial markets.
method Characterizes deterministic strict equilibrium strategies for constant-coefficient and time-invariant probability weighting functions. Addresses the issue of selecting an optimal strategy from multiple equilibrium strategies for time-variant probability weighting functions.
result Characterizes deterministic strict equilibrium strategies and identifies optimal strategies from multiple equilibrium strategies.
The scalar curvature equation for rotation invariant Kähler metrics on Cn\{0} is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on Cn\{0} in lower dimensions. We also prove that there doe…
Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.
problem Stability of the Schwarzschild singularity in near-Schwarzschild black holes.
method Energy methods and new approach to Einstein vacuum equations in axial symmetry.
result The solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity.
Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.
problem Understanding the movement patterns of ants in a 6D space.
method Analyzing mechanical system rules to derive distribution structures and singular trajectories.
result Distributions and singular trajectories of ants' movement rules in a 6D space.
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the first and second derivatives, and with holomorphic coefficients.
The paper proves an equilibrium in a limited stock market participation model with power utilities.
problem Existence of an equilibrium in a model with limited stock market participation and power utilities.
method Proves existence and uniqueness of a solution to a singular and path-dependent Riccati-type ODE.
result Proves existence of a Radner equilibrium with homogenous power-utility investors.
New methods prove existence of rotating shapes moving in space.
problem Existence of rotating shapes moving in space.
method Different methods to prove existence based on singular ordinary differential equation.
result Existence of rotationally symmetric translating solutions proven without partial differential equations.
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Study proves fluid limits of fragmented limit-order markets.
problem Modeling fragmented limit-order markets with small and frequent orders.
method Proved convergence of discrete system to fluid limit characterized by coupled nonlinear ODEs.
result Fluid system converges to stationary equilibrium state over time.
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory a…
Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing f…
This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-sim…
The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mea…
This study shows why training Neural ODEs is hard and proposes a new method.
problem Training Neural ODEs is challenging, especially in practice.
method Proposed a new stabilization method and provided an analytical convergence analysis.
result Insights and techniques for researchers starting work on Neural ODEs.
Time series with non-uniform intervals occur in many applications, and are difficult to model using standard recurrent neural networks (RNNs). We generalize RNNs to have continuous-time hidden dynamics defined by ordinary differential equations (ODEs), a model we call ODE-RNNs. Furthermore, we use ODE-RNNs to replace t…
Proves Hawking's theorem for less smooth spacetime metrics.
problem Proving Hawking's singularity theorem for less smooth spacetime metrics.
method New estimates for Ricci curvature and a segment-type inequality for volume control.
result Proves Hawking's singularity theorem for Lipschitz metrics.
Minimal surfaces in third-order ODEs identified for linear second-order ODEs.
problem Characterizing minimal surfaces in third-order ODEs.
method Analyzing submanifolds of third-order ODEs as Riemannian manifolds.
result Linear second-order ODEs with y′′=±y+β(x) are the only minimal surfaces and totally geodesic. DALTON improves ODE parameter estimation by learning from noisy data.
problem High sensitivity to parameters in ODEs produces unreliable parameter estimates.
method Data-adaptive probabilistic likelihood approximation for ODEs.
result DALTON produces more accurate parameter estimates than existing methods.
The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.
problem Understanding singularities and deformations in meromorphic connections and quadratic differentials.
method Local formal invariants and jets of meromorphic quadratic differentials, universal isomonodromic deformation, unfolded Stokes phenomenon, horizontal and vertical foliations.
result Establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials, describing parameter spaces and moduli spaces.
Improved neural-ODE for faster convergence and stability.
problem Stability, consistency, and convergence issues in neural-ODE solvers.
method Proposed a first-order Nesterov's accelerated gradient (NAG) based ODE-solver.
result Efficacy demonstrated in three tasks: supervised classification, density estimation, and time-series modelling.
New method uses Gaussian ODE filtering to approximate likelihoods for fast ODE inverse problems.
problem Intractable forward models in likelihood-free inference, especially for ODEs.
method Gaussian ODE filtering to construct local Gaussian likelihood approximations.
result New solvers outperform standard likelihood-free approaches on benchmark systems.
Faster training of neural ODEs using Gauß-Legendre quadrature.
problem Training neural ODEs is slow due to solving ODEs numerically.
method Use Gauß-Legendre quadrature to solve integrals faster than ODE-based methods.
result Faster training of neural ODEs, especially for large models.
We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive…
Paper analyzes convergence of ODE samplers in Wasserstein distances.
problem Limited theoretical understanding of convergence properties of probability flow ODEs.
method Convergence analysis for general probability flow ODEs in 2-Wasserstein distance.
result First non-asymptotic convergence analysis for probability flow ODE samplers.
First-order ODEs linked to flat surfaces, leading to integrability.
problem Integrating first-order ODEs.
method Defined Riemannian metrics on variable spaces, studied surface properties, and established connections between Jacobi fields and Lie point symmetries.
result Flat associated surfaces lead to integrable first-order ODEs.
A new Fourier model improves ODE prediction.
problem Improving the accuracy of ODE solutions, especially for periodic functions.
method Constructing a Fourier state space model and a hybrid model combining Taylor and Fourier methods.
result The hybrid model can predict ODE solutions more accurately, especially for periodic functions.
New ODE solvers improve training efficiency and accuracy.
problem Training Neural ODEs requires efficient and accurate gradient calculation.
method Presented algebraically reversible ODE solvers that are time and memory efficient, calculate exact gradients, and are numerically stable.
result Reversible solvers strictly improve upon previous architectures in efficiency and accuracy.
This paper uses ODE to improve RNN models for time series data.
problem Improving RNN models for irregularly sampled time series data.
method Extending RNNs with Neural Ordinary Differential Equations (ODEs).
result New ODE-based RNN models reduce training and evaluation time.
We present Ordinary Differential Equation Variational Auto-Encoder (ODE2VAE), a latent second order ODE model for high-dimensional sequential data. Leveraging the advances in deep generative models, ODE2VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex conti…
New method combines ODE filters and numerical quadrature to propagate model uncertainty.
problem Propagation of model uncertainty in ODE solutions with uncertain parameters.
method Combining ODE filters with numerical quadrature.
result Effective propagation of both numerical and parametric uncertainty.
New symmetry dimensions for higher order ODEs are identified.
problem Determining the maximal and submaximal symmetry dimensions for higher order ODEs.
method Cartan-geometric approach to classify symmetry dimensions.
result Next largest realizable symmetry dimensions for scalar ODEs of order ≥ 4 and vector ODEs of order ≥ 3 are determined.
The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear…
Enhanced Neural ODEs outperform traditional models in image classification and video prediction.
problem Efficiently modeling time-varying dynamics in neural networks.
method Proposed a novel family of non-autonomous Neural ODEs with time-varying weights.
result Outperformed previous Neural ODE variants in speed and representational capacity.
Paper improves neural ODEs for forecasting non-Markovian processes.
problem Forecasting irregularly observed time series with incomplete data.
method Path-dependent Neural Jump ODEs with signature transform.
result Path-dependent NJ-ODE outperforms original framework in non-Markovian data.
This paper explores normalization in neural ODEs, achieving high accuracy in CIFAR-10.
problem Understanding the role of normalization in neural ODEs.
method Investigated different normalization techniques and their impact on neural ODEs performance.
result Achieved 93% accuracy in CIFAR-10 classification task.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Neural ODEs' performance varies with numerical method, requiring adaptive step size control.
problem Neural ODEs' performance depends on the numerical method used during training.
method Proposes an adaptive step size control algorithm to ensure a valid ODE without increasing computational cost.
result Valid Neural ODEs require careful numerical method selection and step size adaptation.
A simple regularization technique speeds up training of Neural ODEs.
problem Training Neural ODEs is computationally expensive.
method Randomly sampling the end time of the ODE during training.
result Significantly decreases training time and improves performance.
New vector fields integrate first-order ODEs.
problem Integrating first-order ODEs.
method Relation between Riemannian manifolds and ODEs integration.
result Integration procedure for first-order ODEs.
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, de…
New method trains neural ODEs faster with fewer layers.
problem Training neural ODEs on large datasets is computationally expensive.
method Combines optimal transport and stability regularizations.
result Significant reductions in training time with no performance loss.
This work improves likelihood of score-based diffusion ODEs using high-order denoising score matching.
problem The gap between maximum likelihood and score matching objectives for score-based diffusion ODEs.
method High-order denoising score matching to maximize likelihood.
result Score-based diffusion ODEs achieve better likelihood on synthetic and CIFAR-10 data.
New method sparsifies hybrid neural ODEs for better performance and stability.
problem Excessive latent states and interactions from mechanistic models lead to training inefficiency and over-fitting.
method Automatic state selection and structure optimization combining domain-informed graph modifications with data-driven regularization.
result Improved predictive performance and robustness with desired sparsity.